|
%I #29 Sep 04 2022 04:06:02
%S 495,11583,135135,1061775,6370650,31286970,131405274,486370170,
%T 1621233900,4946841900,13992495660,37058912748,92647281870,
%U 220089696750,499568676750,1088533853550,2285921092455,4642276728375,9143878404375,17513561154375,32691980821500,59592810754620
%N a(n) = binomial(n+8,8)*binomial(n+12,8).
%C It appears that all terms are multiples of 99. - _Michel Marcus_, Aug 01 2015
%H <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (17,-136,680,-2380,6188,-12376,19448,-24310,24310,-19448,12376,-6188,2380,-680,136,-17,1).
%F a(n) = A000581(n+8)*A000581(n+12). - _Michel Marcus_, Aug 01 2015
%F From _Amiram Eldar_, Sep 04 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 11648*Pi^2/3 - 65726161036/1715175.
%F Sum_{n>=0} (-1)^n/a(n) = 262144*log(2)/99 - 629604992/343035. (End)
%e a(0): C(0+8,8)*C(0+12,8) = C(8,8)*C(12,8) = 1*495 = 495.
%e a(7): C(7+8,8)*C(7+12,8) = C(15,8)*C(19,8) = 6435*75582 = 486370170.
%t f[n_] := Binomial[n + 8, 8] * Binomial[n + 12, 8]; Table[ f[n], {n, 0, 18}] (* _Robert G. Wilson v_, Apr 19 2005 *)
%o (PARI) vector(30, n, n--; binomial(n+8,8)*binomial(n+12,8)) \\ _Michel Marcus_, Jul 31 2015
%o (Magma) [Binomial(n+8,8)*Binomial(n+12,8): n in [0..30]]; // _Vincenzo Librandi_, Jul 31 2015
%Y Cf. A000581.
%K easy,nonn
%O 0,1
%A _Zerinvary Lajos_, Apr 18 2005
%E Corrected and extended by _Robert G. Wilson v_, Apr 19 2005
%E a(6) corrected by _Georg Fischer_, May 08 2021
|
